1D modelling of an alpha type Stirling engine

Int. J. Simul. Multisci. Des. Optim. 2014, 5, A07
N. Martaj, P. Rochelle, Published by EDP Sciences, 2014
DOI: 10.1051/smdo/2013019
Available online at:
www.ijsmdo.org
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1D modelling of an alpha type Stirling engine
N. Martaj1,* and P. Rochelle2
1
2
EPF-Ecole d’Ingénieurs, 21 Bd. Berthelot, 34000 Montpellier, France
Laboratoire d’Energétique, de Mécanique et d’Electromagnétisme, Université Paris Ouest Nanterre-La Défense, 50, rue de Sèvres,
92410 Ville d’Avray, France
Received 10 July 2013 / Accepted 13 November 2013 / Published online 6 February 2014
Abstract – In this paper, a 1-D model of an half alpha type ‘‘double-acting’’ Stirling engine (made up of two doubleacting pistons, two hot heat-exchangers, two cold heat-exchangers and a common recuperator-regenerator for the two
separate gas circuits, all that giving two cycles with 180 phasing) is presented. In this architecture, two cylinders,
containing, each one, a double-acting piston, are combined in order to operate like two parallel Stirling systems in
opposition of phase. This model, derived from Andersen’s model, is used to describe the compressible flows in the
half-engine, under energy transfers. A finite-volume numerical method is applied for the equations of energy, mass
and momentum assessment. This modelling was carried out using the Matlab/Simulink software. The results of this
dynamic modelling relate to one half engine. We obtain the evolution of the physical parameters (density, pressure
and temperature), and the (p, V) cycle. The influence of the various assumptions was studied. A parametric study
was carried out in order to obtain the optimal values of the geometry of the engine and its ideal speed of operation.
Key words: Energetical analysis, Irreversibilities, Optimisation; Stirling engine, Alpha type, Micro solar power plant.
Nomenclature
A
cp
C
D
h
k
L
m
N
P
Q
Q
R
S
T
To
t
V
U
v
W
W
Heat transfer area, m2
Specific heat at constant pressure, J/(kg K)
Stroke, m
Diameter of the piston m
Heat transfer coefficient W/(m2 K)
Friction coefficient
Length, m
Mass, kg
Engine speed, rev/min
Pressure, Pa
Heat flow, W
Heat, J
Gas constant, J/(kg K)
Cross section, m2
Temperature, K
Torque, Nm
Time, s
Volume, m3
Internal energy, J
Speed, m/s
Work, J
Power, W
*e-mail: [email protected]
Greek letters
q
k
l
d
g
Fluid density, kg m3
Thermal conductivity, W/mK
Dynamic viscosity of gas, Pa s
Gap between the cylinder and the segment m
Efficiency [%]
Subscripts
c
e
f
h
k
p
r
w
Compression
Expansion
Friction
Hot
Cold
Operating piston
Regenerator
Wall
1. Introduction
In this study we present the results of a Stirling engine
simulation, part of a relatively low cost and autonomous micro
solar power plant (MiCST) project, based on the use of simple
and robust elements to produce electricity for remote sites in
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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N. Martaj and P. Rochelle: Int. J. Simul. Multisci. Des. Optim. 2014, 5, A07
Solar collectors
Heat storage tank with
exchangers (medium
temperature)
Stirling thermal
engine
Heat
source
Cold
sink
Alternator
& Inverter
Circuits of coolants with isolating valves
Figure 1. Schematic of the micro solar power plant.
Double effect piston
Hot space (expansion)
Cold space (compression)
Recuperator
Exchangers
Figure 2. Double effect alpha type Stirling engine with recuperator.
emerging countries. Solar heating could be realized by
moderate-temperature parabolic-trough receivers. The heat storage could be achieved by sensible heat at moderate temperature
as T = 300 C using materials which will be locally available
(sand, clay, etc.).
To elaborate a dynamic modelling of the Stirling engine of a
micro solar power plant (Figure 1), we built a mono-dimensional model, giving the instantaneous speed and pressure of
the working gas, derived from the Andersen’s one [1].
The used architecture of the Stirling engine is presented in
Figure 2. This alpha type Stirling engine has two cylinders containing, each one, a double effect piston. The two pistons compress the working fluid (air) and receive the mechanical power
on their two faces; this double-acting model, presents a symmetrical configuration around the crankshaft.
The originality of this engine is that the regenerator or recuperator is a common exchanger for the two parallel circuits in
which heat exchanges are done.
This model uses a numerical method of finite volumes. It
uses also the energy and the momentum assessment of the compressible flows, and takes into account the internal irreversibilities, like the pressure losses in the exchangers, and includes the
dependence of the transfer and friction coefficients to the local
Reynolds number [2–4].
For this dynamic modelling of the alpha type Stirling
engine, we use the Matlab/Simulink environment.
The engine is mainly made up of at least five volumes
(Figure 3), but much more subdivided volumes if necessary:
–
–
–
–
–
volume
volume
volume
volume
volume
of
of
of
of
of
expansion (e),
the hot exchanger (h),
heat stocking/destocking (recuperator) (r),
the cold exchanger (k),
compression (c).
2. Mono-dimensional model derived from
Andersen model
This model allows a mono-dimensional description of the
engine: each volume, such as defined previously, is discretized
into one or several main cells. Overlapping these cells, secondary others (with odd numbers) are defined, each one containing
the interface of two successive main cells (Figures 3 and 4)
N. Martaj and P. Rochelle: Int. J. Simul. Multisci. Des. Optim. 2014, 5, A07
3
Figure 3. Subdivision Diagram of the elementary Stirling engine.
The masses related to the secondary cells, including the
interfaces, are calculated as the average of those of the overlapped volumes.
m2n þ m2nþ2
:
ð2Þ
m2nþ1 ¼
2
Figure 4. Representative diagram of 3 successive main volumes of
the Andersen model with secondary volumes (hatched areas
#2n 1 and #2n + 1).
In these volumes, we consider the equations of the mass
and energy conservation, applied to the main cells and the equation of the momentum in the secondary cells. The physical
parameters and sizes can thus be imposed or calculated in each
one of these cells and on their walls.
The notations used in the equations are as follows:
n: sequence number of volumes,
2n: order number of the main cells,
2n + 1: order number of the secondary cells.
For example, in the elementary case, where there is no discretization, there are five main volumes 2n (numbers 2, 4, 6, 8,
10), in Figure 3. Volumes 2 and 10 are, respectively, those of
expansion and compression. Volumes 4 and 8 are respectively
in contact with the hot source and with the cold source. Volume
6 is the recuperator volume.
The equations considered are those of the mass and energy
conservation which are applied to the main cells. The equation
of the momentum is applied to the secondary cells (which
include the junction of the main cells).
2.1. Description of the model
Mass conservation:
The mass assessment for the 2n numbered main volume is
written as follows:
dm2n
ð1Þ
¼ q2nþ1 v2nþ1 S 2nþ1 q2n1 v2n1 S 2n1 :
dt
The speeds v is counted positives when they are oriented in
the order of volumes.
This equation allows, after integration, to obtain the masses
in even volumes (main cells).
Energy conservation:
The first law of thermodynamics, for an opened system,
applied to the 2n numbered volume, gives the variation of its
internal energy U by:
dE2n
dU 2n
¼
¼ q2nþ1 v2nþ1 S 2nþ1 cp2nþ1 T 2nþ1
dt
dt
q2n1 v2n1 S 2n1 cp2n1 T 2n1 g
dV 2n
:
ð3Þ
þ h2n A2n ðT w2n T 2n Þ p2n
dt
For perfect gas, internal energy is a function of the temperature T: U = mÆcvÆT.
We thus obtain the expression of the derivative of the temperature in each volume 2n.
dU 2n dðm2n cv2n T 2n Þ
dT 2n
!
¼
dt
dt
dt
1
dU 2n U 2n dm2n
:
m2n cv2n dt
m2n dt
ð4Þ
The interface temperatures are calculated as follows:
If v2nþ1 0 T 2nþ1 ¼ T 2n ; else T 2nþ1 ¼ T 2nþ2 :
ð5Þ
Momentum equation:
Speeds at the interfaces 2n + 1 are calculated from the
equation of the following momentum assessment:
dv2nþ1
1 n
¼
p2nþ2 S2nþ2 þ p2n S2n
m2nþ1
dt
þ ~p2nþ1 ðS2nþ2 S2n Þ
dm2n
dm2nþ2
ðv2n v2nþ1 Þ ðv2nþ2 v2nþ1 Þ
þ
dt
dt
o
ð6Þ
v2nþ1
Fw2nþ1 þ F AD 2nþ1 ;
jv2nþ1 j
where FADi represents the artificial dissipation force in volume
i [1].
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N. Martaj and P. Rochelle: Int. J. Simul. Multisci. Des. Optim. 2014, 5, A07
The forces due to the pressure losses in the auxiliary cell
2n + 1 are expressed in the following form:
Fw2nþ1 ¼ pf ; 2nþ1 Sref ;2nþ1 ;
ð7Þ
where
S ref ;2nþ1 ¼ minðS2n;2nþ1 ; S2nþ2; 2nþ1 Þ i and pf ; 2nþ1 ¼
h
k f 2n q2n v22n DLh2n2n þ k f 2nþ2 q2nþ2 v22nþ2 DLh2nþ2
with kf is the
2nþ2
friction factor and q the gas density.
The pressure at the interface 2n + 1 can be expressed by:
1
4
~p2nþ1 ¼ p2nþ2 þ
1
2
q2nþ1 v22nþ2 ð1 ¼ p2n þ 12 q2nþ1 v22n ð1 S c;2nþ2
Þ si
S c;2n
S c;2n
Þ si
S c;2nþ2
S c; 2n < S c; 2n þ 2
S c; 2nþ2 S c; 2n
ð8Þ
with Sc,2n and Sc,2n+2 as cross sections of the gas flow, in the
cells 2n and 2n + 2.
Representative gas speeds in the main cells are:
v2n ¼
v2n1 S2n1 þ v2nþ1 S2nþ1
:
2S2n
ð9Þ
The pressures are given here using the differential equation
of perfect gases:
dp2n
1 dm2n
1 dT 2n
1 dV 2n
¼ p2n
þ
ð10Þ
m2n dt
T 2n dt
V 2n dt
dt
and
p2nþ1 ¼
p2n þ p2nþ2
:
2
ð11Þ
2.2. Results of simulation with five main cells
In this first elementary step of simulation, we consider:
j five volumes,
j constant characteristics of working
k = 36.103 W/mK; l = 25.106 Pa s,
j the exchangers used are with plates:
gas
(air):
– width of the plates: 200 mm;
– heat transfer coefficients are, here, supposed constants
[9]: hh = hk = hr = 900 W/m2 K;
– friction coefficient, kf = 0.04 [1, 16].
j compression and expansion cylinder volumes are supposed adiabatic,
j each volume is reduced to a central cell (hot and cold
exchanger, recuperator, cylinder of compression and cylinder of expansion).
Initial pressure is 22.7 bars (calculated from the initial
conditions)
mr
pi ¼ V e V
V ;
Vr
i
h
þ
þ
þ VT kk þ Tcci
Te
Th
Tr
The recuperator temperature is the average of hot and cold temperature sources.
The (p-V) cycles in compression and expansion spaces are
plotted only for the last cycle (diagrams (pe-Ve) and (pc-Vc) for
the 20th cycle) in Figure 5.
The results obtained with this model, are presented in
Table 1.
The indicated power of the engine, calculated by this model is
5602 W for a velocity about 300 rev/min. This indicated power
was calculated by integration on this last cycle. A relative error
of 0.54% was calculated, comparing this value to the power
obtained by the energy balance. The total heat provided to the
cycle is the sum of the heat provided to hot volume Qh and heat
Qr, necessary for the compensation of regeneration losses.
The representative speeds of gas in the main cells depend on
the local density of the gas which evolves with the temperature.
To take into account this dependence, we expressed these speeds
by the following equation, which replaces the equation (9):
v2n ¼
q2n1 v2n1 S2n1 þ q2nþ1 v2nþ1 S2nþ1
:
2q2n S2n
ð12Þ
The new results obtained with the Matlab program are presented in Table 2.
2.3. Effect of the subdivision of the recuperator
on engine operation
In this section, a monodimensional modelling is presented
using several main cells in the regenerator in order to impose
a linear variation of temperature in this exchanger. The results
obtained by subdividing the recuperator/regenerator in five
main cells, using the previous initial data (including the coefficient of transfer of 900 W/m2 K, pressure losses 0.04 and the
speed about 300 rev/min) are represented in Table 3.
In order to analyse the influence of the subdivision of the
engine volumes on its calculated performance, calculations
were made for a recuperator divided initially into 10 then into
20 main cells, maintaining hot and cold volumes subdivided
into five main cells, for a speed of 400 rev/min. The results
obtained for the maximal indicated power Pi, max and the maximal efficiency gi, max are as follows:
j For a 20 cells recuperator Pi,max = 6625 W and
gi,max = 5.4%,
j For a 10 cells recuperator Pi,max = 6562 W and
gi,max = 5.2%.
We notice that there is not a great difference between the
results obtained for these two configurations (10 and 20 cells
at the recuperator). The conclusion is that we can limit a reasonable number of subdivisions (10 in this case) to get faster
numerical simulations with satisfactory results.
2.4 Addition of friction coefficient correlations
on heat exchangers
With the previous initial data, the results are obtained using
constant heat transfer and friction coefficients. But physics constraints and reality tell us to use coefficients related to the
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N. Martaj and P. Rochelle: Int. J. Simul. Multisci. Des. Optim. 2014, 5, A07
5
x 10
6
4.5
Pression en Pa
4
(pe, Ve)
3.5
3
(pc, Vc)
2.5
2
1.5
1
0
1
2
3
4
3
Volume en m
5
6
7
x 10
-3
Figure 5. (p, V) diagrams in expansion and compression volumes.
Table 1. Results of simulation – monodimensional model, five
volumes, one cell in the recuperator.
Table 3. Result of simulation – one-dimensional model, five cells in
the recuperator.
Quantities
Heat received on hot exchanger, Qh, [J]
Variation of heat to the recuperator Qr, [J]
Heat provided to the cold exchanger Qk, [J]
Indicated efficiency [%]
Indicated work [J]
Quantities
Qh, [J]
Qr, [J]
Qk, [J]
Wi, [J]
Pi, [W]
gi, [%]
Values
23,089
2180
19,795
5.36
1120
Table 2. Results of simulation – one-dimensional model, five
volumes, one cell in the recuperator, dependence of v on the gas
density.
Quantities
Heat received on the hot exchanger, Qh, [J]
Variation of heat to the recuperator Qr, [J]
Heat provides to the cold exchanger Qk, [J]
Indicated efficiency [%]
Indicated work [J]
Indicated power [W] on 300 rev/min
Values
16,237
3325
11,776
8.75
1130
5650
Reynolds number of the local flow. The correlations of the Nusselt number and the coefficient of friction, used for the smoothplate heat exchangers [10–15], are:
– For the laminar flow:
Nu ¼ 0:66 and kf ¼ 64=Re;
ð13Þ
– For the turbulent flow:
Nu ¼ 0:02 Re0:8 and k f ¼ 0:32 Re0:25 :
ð14Þ
Values
16,193
3277
11,799
1122
5610
8.69
In these calculations, we considered the following
assumptions:
– five cells in the recuperator,
– one cell for each other volume (hot and cold exchanger,
compression cylinder and expansion cylinder).
Table 4 summarizes the influence of the smooth-plates heat
exchangers on the performances of the engine at a speed of
300 rpm.
The figures which follow were obtained using 10 main cells
for the recuperator and 5 for each exchanger (22 main cells to
represent the engine). The charging pressure is about 10 bars.
We note, in Figures 6–8, the quasi homogeneity of pressure
in the circuit, the strong fluctuations of local temperature, which
are contradictory to the ‘‘isothermal process’’ assumption, and
the speeds of gas synchronism and their homogeneity, except
for the sections near the pistons.
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N. Martaj and P. Rochelle: Int. J. Simul. Multisci. Des. Optim. 2014, 5, A07
Table 4. Results of simulation – one-dimensional model, five cells in
the recuperator exchangers with smooth plates.
Quantities
Qh, [J]
Qr, [J]
Qk, [J]
Wi, [J]
Pi, [W]
gi, [%]
Values
14,168
603
12,836
734
3672
5.41
Figure 8. Evolution of the gas speed according to the angle of
crankshaft and to the cell number.
Figure 6. Evolution of the pressure according to the angle of
crankshaft and to the cell number.
the shift of temperature introduced. We will also study its influence on the performances of the engine.
To take into account the thermal inertia of the solid walls,
we establish the heat balance of an element of volume DVw
(Figure 9).
The Newton’s law allows us to express the heat quantities
exchanged by convection between the element of wall volume
and the external fluid as well as the element of wall volume and
working gas.
dQe2n ¼ hl2n A2n ðTw2n Te2n Þ dt;
ð15Þ
dQ2n ¼ h2n A2n ðTw2n T 2n Þ dt:
ð16Þ
The heat balance for the element of wall volume DVW
allows us to determine the heat stored in the wall:
dQw2n ¼ cpw2n qw Vw2n dTw2n
2n2
¼ dQe2n þ dQ2n kw2n Tw2n Tw
Lx
! nx
Tw2nþ2 Tw2n
A2n1 dt þ kw2n
A2nþ1 dt;
Lx
ð17Þ
nx
where LX is the length of the exchanger and nX the number
of cells per exchanger.
The wall temperature variation is given by:
Figure 7. Evolution of the working gas temperature according to the
angle of crankshaft and to the cell number.
2.5 Addition of the thermal inertia in the solid walls
In the preceding modelling, the wall temperatures of the
heat exchangers and of the recuperator were supposed as constants. We will take into account the inertia of these walls which
results in a delay in the change of the temperature of wall during transitory operations [6, 7]. We will quantify this delay and
dTw2n ¼ dQe2n þ dQ2n
þ kw2n ½
þ
Tw2n Tw2n2
Lx
nx
Tw2nþ2 Tw2n
Lx
nx
Þ A2n1
A2nþ1 dt
Results of simulation for:
– a speed of 300 rpm
– an initial pressure of 22.7 bars
1
: ð18Þ
cpw2n qw Vw2n
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N. Martaj and P. Rochelle: Int. J. Simul. Multisci. Des. Optim. 2014, 5, A07
Figure 9. Exchanger wall cells.
Table 5. Results of simulation – one-dimensional model, 10 cells in
the recuperator.
Quantities
Qh, [J]
Qr, [J]
Qk, [J]
Wi, [J]
Pi, [W]
gi, [%]
Values
16,502
449
15109
944
4820
6.01
– aluminium walls, for a linear k between 273 W/mK at
300 K and 237 W/mK at 500 K
– a mean velocity of coolant of 5 m/s and Nu = 0.023Re0.8
Pr0.4
are presented in Table 5.
These results are obtained with five cells for the each
exchanger and 10 cells for the recuperator.
We plot in the Figures 10–12, the changes of the internal
temperatures of working gas and those of the walls according
to the angle of the crankshaft.
We notice that the temperatures of the walls (in blue) of the
hot and cold exchangers and of the recuperator are sinusoidal
with very low amplitude (approximately 1 C), which validates
the assumption of the constant local temperature of the walls
along the time for a steady-operating point.
The Table 6 gives the results of simulation under the
assumption of the constant wall temperatures in the exchangers.
2.5 Study of the combined influence of the out-ofphasing between the two pistons and the initial
pressure
The obtained results for an half-engine allow to locate some
optimum conditions for operation of this engine.
A parametric study was carried out with three values of out
of phasing and two initial pressures:
– 15 bars and 90,
– 10 bars and 110,
– 10 bars and 120.
We obtain a better efficiency (18%) and a power about
8 kW for an out-of-phasing of 120 (Figure 13). In these conditions, the pressures are not as high in the engine as for the preceding cases, giving a smoother operation.
2.6 Addition of the thermal and mechanical losses
Losses by leakage
The joints are preferably placed at the cold side of the beta
type or gamma Stirling engine to avoid the adverse effects due
to the temperature; in our case, each piston will be equipped
with, at least, one ring. Let us consider a piston surrounded
by such a device. If the difference of the pressures is Dp
between the two sides of the piston, the instantaneous rate of
the mass flow is calculated, such as in a small gap with parallel
walls, as follows [5],
d
m_ l ¼ q
p Dd3 p
12 lðT l Þ Ld
ð19Þ
with Ld, the height of the ring.
These losses are estimated at approximately 1500 W for a
gap of 1/100 mm and a ring height of 1 cm. The leak-flow is
substracted or added to the working gas flow depending on
Dp sign.
d Losses by conduction
These losses are due to a heat transfer by conduction on the
hot side towards the cold one of the engine [8]:
Qcond;i ¼ ki Ai
Th Tk
:
Li
ð20Þ
For a difference between hot and cold sources temperature
equal to 200 C, a length about 0.7 m, a heat transfer coefficient by conduction about 250 W/m.K and a heat transfer area
of (0.0126 m2), these losses are estimated at, approximately,
900 W, in absence of insulation between the exchangers.
d Mechanical losses
As for the thermal losses, the mechanical losses must also
be taken into account to evaluate the power and the total effectiveness of the engine [17–20]. First, friction can occur in the
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N. Martaj and P. Rochelle: Int. J. Simul. Multisci. Des. Optim. 2014, 5, A07
Figure 10. Evolution of the gas and wall temperature in the hot exchanger.
Figure 11. Evolution of the gas and wall temperature in the cold exchanger.
N. Martaj and P. Rochelle: Int. J. Simul. Multisci. Des. Optim. 2014, 5, A07
9
Figure 12. Evolution of the gas and wall temperature in the recuperator.
Table 6. Results with constant temperatures in the exchanger walls.
Quantities
Qh, [J]
Qr, [J]
Qk, [J]
Wi, [J]
Pi, [W]
gi, [%]
Values
14,896
1049
12,786
1067
5334
7.7
term of a friction torque, the expression becomes:
Tomf ¼ af þ bf N
:
1000
ð23Þ
The losses by mechanical friction for this alpha type Stirling
engine, with double-acting piston, and crankshaft-connecting
rod kinematics, are estimated at 2.7 kW.
3. Conclusion
kinematics seals/rings of the piston and the displacer as well as
on the shaft. In addition, the seals of the other moving parts can
lead to great forces of friction.
The calculation of frictions is carried out using the following expression:
W f ¼ pmf :V u
ð21Þ
where, pmf is the average pressure of friction in bar, given
by the following equation:
pmf ¼ a þ b
N
:
1000
ð22Þ
N: engine speed rev/min
a = 0.5 constant in bar,
b = 0.1 constant in (bar · 1000)/(rev/min).
These values correspond appreciably to half of those used
in the reciprocating internal combustion engines. Expressed in
In this paper, an original model of double effect Stirling
engine was studied. Its originality lies in the exchanges in the
regenerator. This is not only a model of a matrix thermal inertia
with back and forth exchanges with the same gas, as in the traditional Stirling engines, but an exchanger with two parallel circuits in which the exchanges are done between two gases
through a wall with some thermal inertia.
We studied a 1-D model, derived from the Andersen model
with finite volumes (volumes are subdivided into cells of calculation). We have introduced variable convective heat transfer
and flow friction coefficients. This model, using mass, momentum and energy balances of the compressible flows, takes into
account the instantaneous internal irreversibilities like the pressure losses in the exchangers and the thermal inertia of the walls
as well as the leakages which were taken into account in the
most advanced calculus. Using this model, we can plot the
speeds, the temperatures, the pressures and the densities of
working gas according to time, and/or angle of crankshaft,
and space (1-D). We obtain the operating characteristics of
the engine (powers, efficiencies).
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N. Martaj and P. Rochelle: Int. J. Simul. Multisci. Des. Optim. 2014, 5, A07
Figure 13. Evolution of the indicated power and the indicated efficiency versus the engine speeds for different phasing: 90, 110 and 120
and initial pressure.
A parametric study showed that a better efficiency (18%)
for a power of approximately 8 kW was obtained for an outof-phasing of 120.
This 1-D model is simulated under Simulink environment
(with multiple cells).
This solar power plant system is under construction and we
have not yet the experimental data at our disposal in order to
compare with simulation results. Our simulation is used to evaluate the system behavior under several conditions and obtain its
power and thermal efficiency and deduce the ‘‘best’’
configuration.
Acknowledgements. The authors gratefully acknowledge Schneider
Electric – leader of MiCST project – and all the partners of the consortium: Barriquand Technologies Thermiques, CEA-INES, Cedrat
Technologies, Defi Systèmes, Exosun, LEME, LEMTA, Mecachrome France, SAED and Stiral. The authors also gratefully
acknowledge the ADEME which greatly supports the MiCST
project.
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Cite this article as: Martaj N & Rochelle P: 1D modelling of an alpha type Stirling engine. Int. J. Simul. Multisci. Des. Optim., 2014,
5, A07.